Chebyshev Polynomials, Rhodonea Curves and Pseudo-Chebyshev Functions. A Survey

Abstract

In recent works, starting from the complex Bernoulli spiral and the Grandi roses, sets of irrational functions have been introduced and studied that extend to the fractional degree the polynomials of Chebyshev of the first, second, third and fourth kind. The functions thus obtained are therefore called pseudo-Chebyshev. This article presents a review of the elementary properties of these functions, with the aim of making the topic accessible to a wider audience of readers. The subject is presented as follows. In Section 2 a review of spiral curves is given. In Section 3 the main properties of the classical Chebyshev polynomials are recalled. The Grandi (Rhodonea) curves and possible extensions are introduced in Section 4, and a method for deriving new curves, changing cartesian into polar coordinates, is touched on. The possibility to consider the Grandi curves even for rational indexes allows to introduce in Section 5 the pseudo-Chebyshev functions, which are derived from the Chebyshev polynomials assuming rational values for their degree. The main properties of these functions are shown, including recursions and differential equations. In particular, the case of half-integer degree is examined in Section 6 since, in this case, the pseudo-Chebyshev functions verify even the orthogonality property. As a consequence, new system of irrational orthogonal functions are introduced.